Consider the subgroup $G_{\lambda}$ of $SL_2(\mathbb R)$ generated by $N_{\lambda} = \begin{bmatrix} 1 & \lambda \\ 0 & 1 \end{bmatrix}$ and $S = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$ where $\lambda>0$. Then it's known by Hecke that $G_{\lambda}$ is discrete if and only if $\lambda \geq 2$ or $\lambda=2 \cos(\frac{\pi}{n})$ where $n \geq 3$ is an integer.
They are named for Hecke, and are used by Hecke to study modular forms.

Is there is a generalization of this fact to more general groups? For example, consider a subgroup of $SL_n(\mathbb R)$ generated by several elementary matrices and some involutions, when is it discrete?

$\begingroup$Even when you consider two unipotent matrices in $SL(2,C)$ this problem does not have a good answer (it is undecidable in certain sense). mathoverflow.net/questions/109967/…$\endgroup$
– MishaFeb 8 at 20:35

$\begingroup$@Misha Thank you for that answer.. How about other groups? For example $SL_3(\mathbb R)$?$\endgroup$
– zzyFeb 11 at 20:58

1

$\begingroup$@zzy: I do not know for sure about $SL(3,R)$. For "positive relatively Anosov" subgroups (a very special class of discrete subgroups) a description should be possible. But for general discrete subgroups I am very skeptical. However, not enough is known about discrete subgroups of $SL(3,R)$ at this point to prove anything definitive.$\endgroup$
– MishaFeb 11 at 21:24